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Properties

Label 27.7.f.a
Level $27$
Weight $7$
Character orbit 27.f
Analytic conductor $6.211$
Analytic rank $0$
Dimension $102$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [27,7,Mod(2,27)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(27, base_ring=CyclotomicField(18))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 7, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("27.2");
 
S:= CuspForms(chi, 7);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 27 = 3^{3} \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 27.f (of order \(18\), degree \(6\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(6.21146025774\)
Analytic rank: \(0\)
Dimension: \(102\)
Relative dimension: \(17\) over \(\Q(\zeta_{18})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{18}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 102 q - 6 q^{2} - 6 q^{3} - 6 q^{4} + 210 q^{5} - 342 q^{6} - 6 q^{7} - 9 q^{8} + 1242 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 102 q - 6 q^{2} - 6 q^{3} - 6 q^{4} + 210 q^{5} - 342 q^{6} - 6 q^{7} - 9 q^{8} + 1242 q^{9} - 3 q^{10} - 492 q^{11} + 3549 q^{12} - 6 q^{13} - 12183 q^{14} - 7974 q^{15} - 198 q^{16} - 9 q^{17} + 40995 q^{18} - 3 q^{19} - 19767 q^{20} - 2406 q^{21} - 10806 q^{22} + 14466 q^{23} + 35352 q^{24} - 15990 q^{25} + 64548 q^{27} - 12 q^{28} - 45690 q^{29} - 186003 q^{30} + 18354 q^{31} - 140544 q^{32} - 78804 q^{33} - 82278 q^{34} + 268263 q^{35} + 412632 q^{36} - 3 q^{37} + 28428 q^{38} + 126402 q^{39} - 34593 q^{40} + 234948 q^{41} - 466074 q^{42} + 148764 q^{43} - 1222785 q^{44} - 646578 q^{45} - 3 q^{46} + 517098 q^{47} + 1608681 q^{48} - 311262 q^{49} + 1710663 q^{50} + 825291 q^{51} - 213057 q^{52} - 1089018 q^{54} - 12 q^{55} - 3167925 q^{56} - 1657317 q^{57} + 663789 q^{58} - 230802 q^{59} - 168786 q^{60} - 51414 q^{61} + 1777536 q^{62} + 1460250 q^{63} + 983037 q^{64} + 2264754 q^{65} + 1054197 q^{66} - 845052 q^{67} - 3979053 q^{68} - 3558150 q^{69} - 835251 q^{70} - 427689 q^{71} - 419292 q^{72} - 257043 q^{73} + 2458185 q^{74} + 3429390 q^{75} + 2759802 q^{76} + 5459610 q^{77} + 5939316 q^{78} + 662682 q^{79} - 3002958 q^{81} - 12 q^{82} - 4926426 q^{83} - 10132716 q^{84} - 2530881 q^{85} - 6700758 q^{86} - 3444642 q^{87} - 2860230 q^{88} + 5907321 q^{89} + 5189148 q^{90} - 451443 q^{91} + 712473 q^{92} - 2952102 q^{93} + 7395807 q^{94} - 2179119 q^{95} + 809838 q^{96} + 272208 q^{97} + 1179270 q^{98} + 1743822 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2.1 −15.2056 − 2.68115i −15.0389 − 22.4239i 163.880 + 59.6475i 37.7314 + 44.9665i 168.554 + 381.289i 398.264 − 144.956i −1476.19 − 852.278i −276.661 + 674.463i −453.165 − 784.905i
2.2 −12.9309 − 2.28007i 19.5181 + 18.6559i 101.870 + 37.0775i −78.6006 − 93.6725i −209.851 − 285.740i 149.932 − 54.5707i −504.967 − 291.543i 32.9160 + 728.257i 802.798 + 1390.49i
2.3 −11.6551 − 2.05510i −14.9225 + 22.5015i 71.4770 + 26.0155i 105.310 + 125.503i 220.166 − 231.590i −278.603 + 101.403i −123.650 − 71.3896i −283.637 − 671.559i −969.470 − 1679.17i
2.4 −9.97059 − 1.75808i 24.1318 − 12.1101i 36.1814 + 13.1690i 60.9980 + 72.6946i −261.899 + 78.3191i −133.791 + 48.6960i 223.553 + 129.069i 435.691 − 584.478i −480.383 − 832.047i
2.5 −9.42516 − 1.66191i −26.9402 − 1.79573i 25.9314 + 9.43824i −132.788 − 158.250i 250.932 + 61.6973i −462.855 + 168.465i 301.732 + 174.205i 722.551 + 96.7550i 988.548 + 1712.22i
2.6 −6.34855 − 1.11942i 2.13616 − 26.9154i −21.0893 − 7.67587i −34.3281 − 40.9106i −43.6912 + 168.482i 90.1091 − 32.7970i 482.595 + 278.626i −719.874 − 114.991i 172.137 + 298.151i
2.7 −3.87880 − 0.683936i −3.37914 + 26.7877i −45.5630 − 16.5836i −43.1394 − 51.4115i 31.4281 − 101.593i 424.254 − 154.416i 383.689 + 221.523i −706.163 − 181.039i 132.167 + 228.919i
2.8 −2.18396 − 0.385091i −26.6089 − 4.57871i −55.5189 − 20.2072i 71.1763 + 84.8246i 56.3496 + 20.2466i 318.690 − 115.994i 236.384 + 136.476i 687.071 + 243.669i −122.781 − 212.663i
2.9 −0.698460 − 0.123157i 20.3194 + 17.7798i −59.6676 − 21.7172i 95.6034 + 113.936i −12.0026 − 14.9210i −285.640 + 103.964i 78.3107 + 45.2127i 96.7599 + 722.550i −52.7432 − 91.3539i
2.10 1.51454 + 0.267054i 26.8676 − 2.67034i −57.9178 − 21.0804i −139.142 − 165.822i 41.4052 + 3.13078i −39.0374 + 14.2084i −167.329 − 96.6072i 714.739 − 143.491i −166.452 − 288.303i
2.11 4.56069 + 0.804173i −8.77856 − 25.5331i −39.9871 − 14.5541i 33.6911 + 40.1515i −19.5033 − 123.508i −487.528 + 177.446i −427.344 − 246.727i −574.874 + 448.287i 121.366 + 210.212i
2.12 6.39689 + 1.12795i −18.1397 + 19.9988i −20.4923 − 7.45860i −37.9291 − 45.2022i −138.595 + 107.470i −230.668 + 83.9562i −482.696 − 278.685i −70.9056 − 725.544i −191.643 − 331.936i
2.13 7.40620 + 1.30591i 20.7548 − 17.2696i −6.99392 − 2.54558i 93.4519 + 111.372i 176.267 − 100.798i 468.870 − 170.655i −465.300 − 268.641i 132.523 − 716.853i 546.682 + 946.881i
2.14 10.4695 + 1.84606i −18.3558 − 19.8006i 46.0625 + 16.7654i −137.909 − 164.354i −155.623 − 241.189i 506.532 − 184.363i −137.929 − 79.6334i −55.1285 + 726.913i −1140.44 − 1975.29i
2.15 11.6192 + 2.04878i 14.4502 + 22.8077i 70.6679 + 25.7210i 2.84045 + 3.38512i 121.171 + 294.613i 129.733 − 47.2189i 114.471 + 66.0899i −311.386 + 659.151i 26.0684 + 45.1518i
2.16 13.7878 + 2.43117i −26.9804 + 1.02752i 124.053 + 45.1517i 118.006 + 140.634i −374.500 − 51.4266i −10.3283 + 3.75918i 824.665 + 476.120i 726.888 − 55.4459i 1285.14 + 2225.93i
2.17 14.6025 + 2.57482i 19.5272 − 18.6464i 146.463 + 53.3083i −47.4564 − 56.5563i 333.158 − 222.005i −559.698 + 203.713i 1179.64 + 681.064i 33.6255 − 728.224i −547.361 − 948.057i
5.1 −9.07449 − 10.8146i −10.8701 − 24.7152i −23.4948 + 133.245i 72.7572 + 199.899i −168.644 + 341.833i −62.5083 − 354.502i 871.728 − 503.293i −492.683 + 537.312i 1501.58 − 2600.82i
5.2 −8.78575 − 10.4705i 21.2323 − 16.6790i −21.3274 + 120.954i −83.1319 − 228.403i −361.178 − 75.7744i 3.85901 + 21.8856i 696.250 − 401.980i 172.622 − 708.267i −1661.11 + 2877.12i
5.3 −8.36754 − 9.97205i −20.7807 + 17.2384i −18.3125 + 103.855i −10.7822 − 29.6238i 345.786 + 62.9829i 45.8281 + 259.904i 467.375 − 269.839i 134.674 − 716.452i −205.189 + 355.398i
See next 80 embeddings (of 102 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2.17
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
27.f odd 18 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 27.7.f.a ✓ 102
3.b odd 2 1 81.7.f.a 102
27.e even 9 1 81.7.f.a 102
27.f odd 18 1 inner 27.7.f.a ✓ 102
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
27.7.f.a ✓ 102 1.a even 1 1 trivial
27.7.f.a ✓ 102 27.f odd 18 1 inner
81.7.f.a 102 3.b odd 2 1
81.7.f.a 102 27.e even 9 1

Hecke kernels

This newform subspace is the entire newspace \(S_{7}^{\mathrm{new}}(27, [\chi])\).